46-944: Stochastic Calculus for Finance 1

Spring 2020

Course Description

The first half of this course introduces martingales, Brownian motion, Itô integrals and Itô’s formula, in both the uni-variate and multi-variate case. This is done within the context of the Black-Scholes option pricing model and includes a detailed examination of this model. The second half of the introduces exponential martingales and the Girsanov theorem. This is done in the context of risk neutral measures and the fundamental theorems of asset pricing.

Learning Objectives

• Develop an understanding of and familiarity with the mathematical tools used to price derivative securities.

Prerequisites

• 46-921: Introduction to Probability
• 46-941: Multi-Period Asset Pricing

References

• Brief lecture notes. (A tablet friendly version is here, and the full TeX source is here.)
• Stochastic Calculus for Finance II by Steven Shreve. (We will cover roughly the first five chapters.)

Class Policies

Lectures

• If you must sleep, don’t snore!
• Be courteous when you use mobile devices.
• Attendance Requirement: The steering committee has requested attendance be recorded and made a part of your grade. Accordingly, attendance will count as 5% of your overall grade, and will be computed as follows:
  • To account for interviews and other special circumstances, you may miss up to 4 lectures without penalty.
  • Missing more than 4 lectures will decrease the attendance portion of your grade proportionally.
  • I will only consider making exceptions to this policy for unexpected severe emergencies that require your absence for more than 12 days.

Homework

• Homework is due at the beginning of class on the due date.
• Late homework policy: Homework turned in within 48 hours of the deadline will be accepted.
  • Late homework turned in within the first 24 hours of the deadline will receive a 10% penalty. In particular, homework turned after class starts, will receive this penalty.
  • Late homework turned in within the next 24 hours will receive a 20% penalty.
  • Homework more than 48 hours late will not be accepted.
  • To account for unusual circumstances, your lowest homework will not count towards your grade.
  • I will only consider making exceptions to the late homework policy for unexpected severe emergencies that require your absence for more than 12 days. Following MSCF policy, I will not make exceptions on account of job interviews or career fares.
• Solutions will usually be posted 48 hours after the homework deadline. Due to holidays or exams solutions to some assignments might post earlier. In this case, you will be notified of this in advance, and late homework will not be accepted after solutions have been posted.
• Your homework must be uploaded as a PDF on Canvas. If you use pencil and paper to write your homework, then you must scan and upload it. Please ensure your scans are a high quality PDF, as photos of your homework will not be accepted.
• You may collaborate on the homework, however, you may only turn in solutions which you fully understand and have written up independently. Violation of this policy will be treated seriously according to procedures in the MSCF student handbook.

Exams

• No notes, calculators, computational aids, or internet enabled devices are allowed during exams.
• You may not give or receive assistance during exams.
• Violation of these policies will be treated seriously according to procedures in the MSCF student handbook.
• If you earn a C+ or lower, and have at least a 70% average on the homework, you may take the makeup final. Your performance on the makeup final can not reduce your grade, and can only increase your grade in the course by at most a full letter, up to a maximum of B–.
• Regrading Policy:
  • Your graded exams will be with your TA for Pittsburgh students, and in the MSCF office for NY students. You may view them in the respective office. However if you take them out of the office you may not request regrading of any problems.
  • If you believe a particular question has been graded incorrectly, then you must do so in writing by leaving a post-it note on the front of the exam indicating which question you want re-graded. Please do NOT include any explanation or message. Your grade will be solely based on our interpretation of what is written on the exam, and not on any explanation you provide outside the exam.
  • We will cover up the original grade, and independently regrade the requested question. The new grade will replace your old grade, even if it is lower. I strongly recommend you read and understand the solutions posted online before requesting a regrade, because your grade could become lower.

Grading

• Attendance will count as 5% of your grade, and homework will count as 10%.
• The remainder 85% of your grade will be determined by your midterm and final, as the higher of:
  • Final 60% and midterm 25%.
  • or Final 85%, reduced by one full letter.
• That is, if you miss the midterm, I will count your Final as 85%, and assess a full letter grade penalty.
• Explicitly, your grade will be computed as follows:
  • Your performance on the homework, midterm and final will each be converted to a numerical grade between 0 and 4.5 “using a curve”.
  • If $F$, $M$, $H$ and $A$ are your numerical grades on the final, midterm, homework and attendance respectively, then your overall grade $G$ will be computed by $$ G = .05A + .1H + \max( 0.6 F + 0.25 M, 0.85 (F - 1) ) \,. $$
  • Your final letter grade will be computed from your numerical grade using the standard scale.

Accommodations for Students with Disabilities

If you have a disability and have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate. If you suspect that you may have a disability and would benefit from accommodations but are not yet registered with the Office of Disability Resources, I encourage you to contact them at access@andrew.cmu.edu.

Student Wellness

As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus here. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.

Faculty Course Evaluations

At the end of the semester, you will be asked to fill out faculty course evaluations. Please fill these in promptly, I value your feedback. As incentive, if over 75% of you have filled out evaluations on the last day of class, then I will release your grades as soon as they are available. If not, I will release your grades at the very end of the grading period.